Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / A / 1 / iv / Solution

The ratio can be evaluated directly:The assumption $|f-g|\to 0$ gives $f-g\to 0$, so continuity of the exponential function givesThe additional premise $f\sim g$ is not needed for this implication. The useful requirement is additive control of the exponents, as explained by asymptotic comparison of exponential functions.